Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.8 → 3.7

Time: 28.3s

Precision: binary64

Cost: 51008

\[z > 0.5\]

Math

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

↓

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{676.5203681218851}{z} + t_0\\ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\frac{\frac{458891030.96596426}{{\left(2 + z\right)}^{3}} + {t_1}^{3}}{\mathsf{fma}\left(t_1, \frac{676.5203681218851}{z} + \left(t_0 + \frac{-771.3234287776531}{2 + z}\right), \frac{\frac{594939.8317813153}{2 + z}}{2 + z}\right)} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5)))
   (exp (- (+ (+ (- z 1.0) 7.0) 0.5))))
  (+
   (+
    (+
     (+
      (+
       (+
        (+
         (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0)))
         (/ -1259.1392167224028 (+ (- z 1.0) 2.0)))
        (/ 771.3234287776531 (+ (- z 1.0) 3.0)))
       (/ -176.6150291621406 (+ (- z 1.0) 4.0)))
      (/ 12.507343278686905 (+ (- z 1.0) 5.0)))
     (/ -0.13857109526572012 (+ (- z 1.0) 6.0)))
    (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0)))
   (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ -1259.1392167224028 (+ z 1.0)))
        (t_1 (+ (/ 676.5203681218851 z) t_0)))
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (+ z 6.5) (+ z -0.5))
     (*
      (exp (- -6.5 z))
      (+
       0.9999999999998099
       (+
        (/
         (+ (/ 458891030.96596426 (pow (+ 2.0 z) 3.0)) (pow t_1 3.0))
         (fma
          t_1
          (+ (/ 676.5203681218851 z) (+ t_0 (/ -771.3234287776531 (+ 2.0 z))))
          (/ (/ 594939.8317813153 (+ 2.0 z)) (+ 2.0 z))))
        (+
         (+
          (/ -0.13857109526572012 (+ z 5.0))
          (/ 9.984369578019572e-6 (- z -6.0)))
         (+
          (+ (/ -176.6150291621406 (+ z 3.0)) (/ 12.507343278686905 (+ z 4.0)))
          (/ 1.5056327351493116e-7 (+ z 7.0)))))))))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
}

↓

double code(double z) {
	double t_0 = -1259.1392167224028 / (z + 1.0);
	double t_1 = (676.5203681218851 / z) + t_0;
	return sqrt((((double) M_PI) * 2.0)) * (pow((z + 6.5), (z + -0.5)) * (exp((-6.5 - z)) * (0.9999999999998099 + ((((458891030.96596426 / pow((2.0 + z), 3.0)) + pow(t_1, 3.0)) / fma(t_1, ((676.5203681218851 / z) + (t_0 + (-771.3234287776531 / (2.0 + z)))), ((594939.8317813153 / (2.0 + z)) / (2.0 + z)))) + (((-0.13857109526572012 / (z + 5.0)) + (9.984369578019572e-6 / (z - -6.0))) + (((-176.6150291621406 / (z + 3.0)) + (12.507343278686905 / (z + 4.0))) + (1.5056327351493116e-7 / (z + 7.0))))))));
}

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5) ^ Float64(Float64(z - 1.0) + 0.5))) * exp(Float64(-Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(z - 1.0) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(z - 1.0) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(z - 1.0) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(z - 1.0) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(z - 1.0) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(z - 1.0) + 6.0))) + Float64(9.984369578019572e-6 / Float64(Float64(z - 1.0) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(z - 1.0) + 8.0))))
end

↓

function code(z)
	t_0 = Float64(-1259.1392167224028 / Float64(z + 1.0))
	t_1 = Float64(Float64(676.5203681218851 / z) + t_0)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(exp(Float64(-6.5 - z)) * Float64(0.9999999999998099 + Float64(Float64(Float64(Float64(458891030.96596426 / (Float64(2.0 + z) ^ 3.0)) + (t_1 ^ 3.0)) / fma(t_1, Float64(Float64(676.5203681218851 / z) + Float64(t_0 + Float64(-771.3234287776531 / Float64(2.0 + z)))), Float64(Float64(594939.8317813153 / Float64(2.0 + z)) / Float64(2.0 + z)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(z + 5.0)) + Float64(9.984369578019572e-6 / Float64(z - -6.0))) + Float64(Float64(Float64(-176.6150291621406 / Float64(z + 3.0)) + Float64(12.507343278686905 / Float64(z + 4.0))) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))))))))
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(z - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(z - 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z - 1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z - 1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z - 1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z - 1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[z_] := Block[{t$95$0 = N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(676.5203681218851 / z), $MachinePrecision] + t$95$0), $MachinePrecision]}, N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] * N[(0.9999999999998099 + N[(N[(N[(N[(458891030.96596426 / N[Power[N[(2.0 + z), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(t$95$0 + N[(-771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(594939.8317813153 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision] / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 3.8

   \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

2. Simplified 3.8

   \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)} \]
   Proof

   [Start] 3.8	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   associate-*l* [=>] 3.8	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   associate-*l* [=>] 3.8	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

3. Applied egg-rr 3.8

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{{\left(\frac{771.3234287776531}{z + 2}\right)}^{3} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2} + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2} \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

4. Simplified 3.7

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{\frac{458891030.96596426}{{\left(z + 2\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}, \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} - \frac{771.3234287776531}{z + 2}\right), \frac{\frac{594939.8317813153}{z + 2}}{z + 2}\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{{\left(\frac{771.3234287776531}{z + 2}\right)}^{3} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2} + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2} \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   cube-div [=>] 3.7	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\color{blue}{\frac{{771.3234287776531}^{3}}{{\left(z + 2\right)}^{3}}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2} + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2} \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\color{blue}{458891030.96596426}}{{\left(z + 2\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2} + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2} \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   +-commutative [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\frac{458891030.96596426}{{\left(z + 2\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\color{blue}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2} \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2}}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   distribute-rgt-out-- [=>] 3.7	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\frac{458891030.96596426}{{\left(z + 2\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\color{blue}{\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2}\right)} + \frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   fma-def [=>] 3.7	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\frac{458891030.96596426}{{\left(z + 2\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) - \frac{771.3234287776531}{z + 2}, \frac{771.3234287776531}{z + 2} \cdot \frac{771.3234287776531}{z + 2}\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

5. Applied egg-rr 3.7

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \color{blue}{{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\frac{\frac{458891030.96596426}{{\left(z + 2\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}, \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} - \frac{771.3234287776531}{z + 2}\right), \frac{\frac{594939.8317813153}{z + 2}}{z + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)}^{1}} \]

6. Final simplification 3.7

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\frac{\frac{458891030.96596426}{{\left(2 + z\right)}^{3}} + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}, \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{-771.3234287776531}{2 + z}\right), \frac{\frac{594939.8317813153}{2 + z}}{2 + z}\right)} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]

Alternatives

Alternative 1
Error	3.7
Cost	51008

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{676.5203681218851}{z} + t_0\\ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\frac{458891030.96596426}{{\left(2 + z\right)}^{3}} + {t_1}^{3}}{\mathsf{fma}\left(t_1, t_0 + \left(\frac{676.5203681218851}{z} + \frac{-771.3234287776531}{2 + z}\right), \frac{594939.8317813153}{\left(2 + z\right) \cdot \left(2 + z\right)}\right)} + \left(\left(0.9999999999998099 + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{12.507343278686905}{z + 4}\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right) \end{array} \]

Alternative 2
Error	3.7
Cost	51008

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{676.5203681218851}{z} + t_0\\ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\frac{\frac{458891030.96596426}{{\left(2 + z\right)}^{3}} + {t_1}^{3}}{\mathsf{fma}\left(t_1, \frac{676.5203681218851}{z} + \left(t_0 + \frac{-771.3234287776531}{2 + z}\right), \frac{594939.8317813153}{\left(2 + z\right) \cdot \left(2 + z\right)}\right)} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)\right) \end{array} \]

Alternative 3
Error	3.7
Cost	45824

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{676.5203681218851}{z} + t_0\\ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\frac{458891030.96596426}{{\left(2 + z\right)}^{3}} + t_1 \cdot \left(t_1 \cdot t_1\right)}{\mathsf{fma}\left(t_1, \frac{676.5203681218851}{z} + \left(t_0 + \frac{-771.3234287776531}{2 + z}\right), \frac{\frac{594939.8317813153}{2 + z}}{2 + z}\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right)\right)\right)\right)\right) \end{array} \]

Alternative 4
Error	3.8
Cost	37824

\[\begin{array}{l} t_0 := \frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\\ \left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(0.9999999999998099 + \frac{\frac{\frac{594939.8317813153}{2 + z}}{2 + z} - t_0 \cdot t_0}{\frac{771.3234287776531}{2 + z} - t_0}\right)\right)\right)\right) \end{array} \]

Alternative 5
Error	3.8
Cost	30912

\[\begin{array}{l} t_0 := 7 + \left(z + -1\right)\\ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(t_0 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(-7 - \left(z + -1\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \frac{771.3234287776531}{3 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{4 + \left(z + -1\right)}\right) + \frac{12.507343278686905}{5 + \left(z + -1\right)}\right) + \frac{-0.13857109526572012}{6 + \left(z + -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \end{array} \]

Alternative 6
Error	3.8
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right)\right)\right) \]

Alternative 7
Error	3.8
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right) \]

Alternative 8
Error	3.8
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right)\right)\right)\right) \]

Alternative 9
Error	46.8
Cost	28992

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 + \left(-6 - z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \frac{246.3374466535184}{z \cdot z}\right)\right)\right)\right) \]

Alternative 10
Error	46.8
Cost	28736

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \frac{246.3374466535184}{z \cdot z}\right)\right)\right)\right) \]

Alternative 11
Error	46.8
Cost	28736

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \frac{\frac{246.3374466535184}{z}}{z}\right)\right)\right)\right) \]

Alternative 12
Error	47.6
Cost	27200

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(0.9999999999998099 + \left(\frac{197.000868054939}{z \cdot z} + \frac{24.458333333348836}{z}\right)\right)\right) \]

Alternative 13
Error	51.4
Cost	26816

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right) \]

Alternative 14
Error	51.6
Cost	26756

\[\begin{array}{l} \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 15
Error	52.0
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 16
Error	55.6
Cost	19712

\[\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)} \]

Reproduce

herbie shell --seed 2023012 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  :pre (> z 0.5)
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))